The determinant of one-dimensional polyharmonic operators of arbitrary order

We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator Pn=(−1)n(∂x)2n on (0,T) with Dirichlet boundary conditions and n a positive integer, and show that it satisfies the asymptotics log⁡(det⁡Pn)=−n2log⁡n+[7ζ(3)2π2+32+log⁡(T4)]n2+O(n) for large n. This is a consequence of sharp upper and lower bounds for log⁡(det⁡Pn) valid for all n and which coincide in the terms up to order n. These results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour.